We can now move to analyse in more detail the various cases.
First of all we consider the ungroomed \(\lambda ^1_{1/2}\)  distribution for the lower transverse momentum bin in Fig. REF .
From the qualitative argument of Eq. (REF ), we expect the boundary between the first two regions to be located
around \(\lambda ^1_{1/2} \simeq 0.1\) . Indeed, in the region to the left of this value the agreement between the \(\text{NLO} +\text{NLL}  ^\prime \)  and the three
parton-level MC predictions is rather poor, with the \(\text{NLO} +\text{NLL}  ^\prime \)  and SHERPA MEPS@NLO uncertainty bands not overlapping. Above this value,
the agreement improves, as expected, until we end up in the third region, close to the endpoint, where the predictions from the two approaches
are in strong disagreement.
The situation does not improve with SoftDrop , as we have anticipated earlier. If anything the agreement between the two predictions
deteriorates at the boundary between the first and second region. We
attribute this to the presence of transition-point effects (remember
that we have chosen \(z_{\rm cut}=0.1\) ). However, as these are formally NNLL effects,
they are not very-well modelled by our calculation.
In the case of the higher transverse momentum bin, the boundary between the first two regions is pushed to smaller values of the observable
(\(\lambda ^1_{1/2} \simeq 0.05\) ) and, consequently, the region where we find agreement between resummation and MC simulations should widen.
This appears clearly in the top-left plot of Fig. REF , which represents the ungroomed case. In fact, the
\(\text{NLO} +\text{NLL}  ^\prime \)  result is in very good agreement with the PYTHIA prediction. Changes in behaviour of the groomed distribution with respect to the
lower-\(p_T\)  case are instead minimal.
In both cases, we still see large deviations in the endpoint region.
Therefore, we must conclude that the distribution of the LHA observable is not
well theoretically controlled in our \(\text{NLO} +\text{NLL}  ^\prime \)  calculation. Since the different
parton-shower simulations agree among themselves much better, a more
systematic study of the differences with respect to the analytic approach
seems well motivated. In this context we note that Ref. [1]}
presented such comparison for the closely related BKS observables
[2]}, [3]} and fractional-energy correlations
[4]} in \(e^+e^-\)  collisions, in an approach that could be
extended to groomed observables, also in hadronic collision. In any case,
as we shall see shortly, the other angularities considered in this study do
not suffer from this problem to the same extend.
